If ` x = y + z then x^(3) - y^(3) - z^(3)` is

Options

A. 0
B. 3xyz
C. `-3xyz`
D. 1

Answer

To solve the expression \( x^3 - y^3 - z^3 \) given that \( x = y + z \), we can follow these steps:

### Step 1: Substitute the value of \( x \)
Since we know that \( x = y + z \), we can substitute this into the expression:
\[ x^3 - y^3 - z^3 = (y + z)^3 - y^3 - z^3 \]

### Step 2: Expand \( (y + z)^3 \)
Using the binomial theorem, we can expand \( (y + z)^3 \):
\[ (y + z)^3 = y^3 + 3y^2z + 3yz^2 + z^3 \]

### Step 3: Substitute the expanded form back into the expression
Now, we substitute the expanded form back into our expression:
\[ (y + z)^3 - y^3 - z^3 = (y^3 + 3y^2z + 3yz^2 + z^3) - y^3 - z^3 \]

### Step 4: Simplify the expression
When we simplify the expression, the \( y^3 \) and \( z^3 \) terms cancel out:
\[ (y^3 + 3y^2z + 3yz^2 + z^3) - y^3 - z^3 = 3y^2z + 3yz^2 \]

### Step 5: Factor out the common term
We can factor out \( 3yz \) from the remaining terms:
\[ 3y^2z + 3yz^2 = 3yz(y + z) \]

### Final Result
Thus, the expression \( x^3 - y^3 - z^3 \) simplifies to:
\[ x^3 - y^3 - z^3 = 3yz(y + z) \]